CTET

http://ctet.nic.in/ctetapp/eligibility.aspx
Here is the link where you will get all the information you need about the eligibility criteria of CTET.

No, CTET requires passed or appearing in either Diploma or full degree in teaching education.

Yes, you can give the exam. Please refer the link to understand the eligibility criteria for the exam:
http://ctet.nic.in/CMS/Public/View.aspx?page=39

Yes, you can write the paper in Tamil as well.

When the CTET Forms of Application Open, you should be having your B.Ed. As the Forms come generally when the B.Ed Results have come out, you can still find out this from their websites where they specify that the submission of the B.Ed result is mandatory. Being in the Court Semester in June means that either you are just waiting for the results of 4th Sem. So I am of the view that you can apply with a provisional certificate of Appearing in the B.Ed.

Here is the official link for eligibility criteria:
http://ctet.nic.in/CMS/Public/View.aspx?page=39
visit the same for your query.

Yes, you are eligible for taking the exam. You can fill the application form. Good luck.

No, you need to complete your graduation first as it is the minimum eligibility criteria for TET.

Yes, you can apply for CTET.